Modular curves of prime-power level
with infinitely many rational points
Andrew V. Sutherland
Massachusetts Institute of Technology
March 6, 2016
joint work with David Zywina (Cornell)
Galois representations
Let E be an elliptic curve over Q and let N > 1 be an integer.
The Galois group Gal(Q/Q) acts on the N-torsion subgroup of E(Q),
E[N] ' Z/NZ ⊕ Z/NZ,
via its action on points (coordinate-wise). This yields a representation
ρE,N : Gal(Q/Q) → Aut(E[N]) ' GL2 (Z/NZ),
whose image we denote GE (N). Choosing bases compatibly, we can
take the inverse limit and obtain a single representation
ρE : Gal(Q/Q) → lim GL2 (Z/NZ) ' GL2 (Ẑ),
←−
N
whose image we denote GE , with projections GE → GE (N) for each N.
Modular curves
Let FN := Q(ζn )(X(N)). Then F1 = Q(j) and FN /Q(j) is Galois with
Gal(FN /Q(j)) ' GL2 (Z/NZ)/{±I}
Let G ⊆ GL2 (Z/NZ) be a group containing −I with det(G) = (Z/NZ)× .
Define XG /Q to be the smooth projective curve with function field FNG .
Let JG : XG → X(1) = Q(j) be the map corresponding to Q(j) ⊆ FNG .
If M|N and G is the full inverse image of H ⊆ GL2 (Z/MZ), then
XG = XH . We call the least such M the level of G and XG .
Better: identify G with π−1
N (G), where πN : GL2 (Ẑ) → GL2 (Z/NZ);
G as an open subgroup of GL2 (Ẑ) containing −I with det(G) = Ẑ× .
For any E/Q with j(E) 6∈ {0, 1728}, up to GL2 (Ẑ)-conjugacy,
GE ⊆ G ⇐⇒ j(E) ∈ JG (XG (Q)).
Congruence subgroups
For G ⊆ GL2 (Ẑ) of level N as above, let Γ ⊆ SL2 (Z) be the preimage
of πN (G) ∩ SL2 (Z/NZ).
Then Γ is a congruence subgroup containing Γ (N), and the modular
curve XΓ := Γ \h∗ is isomorphic to the base change of XG to Q(ζn ).
The genus g of XG and XΓ must coincide, but their levels need not (!);
the level M of XΓ may strictly divide the level N of XG .
For each g > 0 we have g(XΓ ) = g for only finitely many XΓ ;
for g 6 24 these Γ can be found in the tables of Cummins and Pauli.
But we may have g(XG ) = g for infinitely many XG (!)
Call g(XG ) the genus of G.
Restricting the level
For each odd prime p there is a G ⊆ GL2 (Z/2pZ) of index 2 that
surjects on to both GL2 (Z/2Z) and GL2 (Z/pZ).
The corresponding XG are all genus 0 curves isomorphic to P1Q with
2
maps JG : XG → X(1) given by JG (t) = 1728 + −1
p pt .
One can similarly construct infinite families of genus 0 curves
XG ' P1Q of level N divisible by any of 3, 5, 7, 13.
And one can take combinations. For example, there are 12
non-conjugate G of level 91 and index at least 24 for which XG ' P1Q .
So we want to restrict N.
Restricting the level
For each odd prime p there is a G ⊆ GL2 (Z/2pZ) of index 2 that
surjects on to both GL2 (Z/2Z) and GL2 (Z/pZ).
The corresponding XG are all genus 0 curves isomorphic to P1Q with
2
maps JG : XG → X(1) given by JG (t) = 1728 + −1
p pt .
One can similarly construct infinite families of genus 0 curves
XG ' P1Q of level N divisible by any of 3, 5, 7, 13.
And one can take combinations. For example, there are 12
non-conjugate G of level 91 and index at least 24 for which XG ' P1Q .
So we want to restrict N.
Theorem: For prime N, there are 29 XG with XG (Q) infinite.
Conjecture: For prime N, there are 64 XG with XG (Q) 6= ∅.
Now let N be a prime power.
Main results
Theorem
There are 248 modular curves XG of prime power level with XG (Q)
infinite. Of these, 220 have genus 0 and 28 have genus 1.
For each of these 248 groups G we have an explicit JG : XG → X(1).
The 201 of 2-power level were independently found by Jeremy Rouse
and David Zureick-Brown [RZB15].1
Corollary
For each of these G we can completely describe the set of j-invariants
of elliptic curves E/Q for which GE ⊆ G.
Corollary
There are 1294 non-conjugate open subgroups of GL2 (Ẑ) of prime
power level that occur as GE for infinitely many E/Q with distinct j(E).
1 Note:
our G act on the left and are thus transposed relative to those in [RZB15].
Finiteness results
Call an open subgroup G ⊆ GL2 (Ẑ) admissible if
I
det(G) = Ẑ× and −I ∈ G;
I
G contains an element GL2 (Ẑ)-conjugate to
1 0
0 −1
or
1 1
0 −1
.
If GE ⊆ G for some E/Q then G must be admissible.
Proposition
For any fixed g > 0, only finitely many admissible G have genus 6 g
and prime power level.
Proposition
There are 220 admissible G of prime power level and genus 0.
There are 250 admissible G of prime power level and genus 1.
Distinguishing finite from infinite when g = 0
Let G ⊆ GL2 (Ẑ) be admissible of `-power level and genus g = 0.
Since g = 0 it is enough to show XG (Q) 6= ∅.
Distinguishing finite from infinite when g = 0
Let G ⊆ GL2 (Ẑ) be admissible of `-power level and genus g = 0.
Since g = 0 it is enough to show XG (Q) 6= ∅.
But this is necessarily true!
Our criterion for admissibility guarantees XG (R) 6= ∅.
For p - ` we must have XG (Qp ) 6= ∅, since X G (Fp ) 6= ∅ (by Hensel).
Thus XG (Q` ) 6= ∅ (by Hilbert).
Therefore XG (Q) 6= ∅ (by Hasse).
Distinguishing finite from infinite when g = 1
If g = 1 then either XG (Q) = ∅, or XG (Q) can be identified with an
elliptic curve EG whose conductor is a power of `.
There are only finitely many such EG , all in Cremona’s tables.
For a finite set S of small primes p - `, their ap -values distinguish
their isogeny classes, and therefore their ranks.
For each of these p - ` we compute ap = p + 1 − #XG (Fp ) where
X
#XG (Fp ) =
#{P ∈ YG (Fp ) : JG (P) = j} + #XG∞ (Fp )
j∈Fp
We compute the sum using the moduli interpretation of YG /Fp .
We don’t need a model for XG , we only need G.
If no match, XG (Q) = ∅. If rank 0 match, XG (Q) is finite.
If rank > 0 match, either XG (Q) = ∅ or XG (Q) is infinite.
To distinguish which applies it suffices to find E/Q with GE ⊆ G.
We find that only 28 of the 250 are elliptic curves of rank r > 0.
#
group
i
N
0
1A0 -1a
3A0 -3a
1
1
3
4
3
3B0 -3a
3C0 -3a
6
3
[0, 1, 2, 0], [1, 0, 0, 2]
(t − 9)(t + 3) / t
1
4
3D0 -3a
12
3
[2, 0, 0, 2], [1, 0, 0, 2]
729 / (t3 − 27)
2
5
9A0 -9a
9B0 -9a
9
9
[0, 2, 4, 0], [1, 1, 4, 5], [1, 0, 0, 2]
−27(t − 3) / (t2 + 3t + 9)
t3 + 9t − 6
1
1
2
6
7
8
9C0 -9a
9D0 -9a
generators
map
3
[0, 1, 2, 0], [1, 1, 1, 2], [1, 0, 0, 2]
t3
0
3
[0, 1, 2, 1], [1, 2, 0, 2]
(t + 3)3 (t + 27) / t
0
12
9
[1, 1, 0, 1], [2, 0, 0, 5], [1, 0, 0, 2]
t(t2 + 9t + 27)
2
9
[2, 0, 0, 5], [4, 2, 3, 4], [1, 0, 0, 2]
t3
2
18
9
[2, 0, 0, 5], [1, 3, 3, 1], [0, 2, 4, 0], [1, 0, 0, 2]
−27 / t3
(t2 − 3) / t
5
9
9E0 -9a
18
9
[1, 3, 0, 1], [2, 1, 1, 1], [4, 2, 0, 5]
9F0 -9a
27
9
[0, 2, 4, 1], [4, 3, 5, 4], [4, 5, 0, 5]
11
9G0 -9a
27
9
[0, 4, 2, 3], [5, 1, 1, 4], [5, 3, 0, 4]
12
9H0 -9a
36
9
13
9H0 -9b
9H0 -9c
36
36
1
[1, 3, 0, 1], [5, 0, 3, 2], [1, 0, 2, 2]
3(t3 + 9)/t3
3t / (2t2 − 3t + 6)
4
9
[1, 3, 0, 1], [5, 0, 3, 2], [2, 1, 0, 1]
9
[1, 3, 0, 1], [5, 0, 3, 2], [4, 2, 0, 5]
3(t3 + 9t2 − 9t − 9) / (t3 − 9t2 − 9t + 9)
−6(t3 − 9t) / (t3 + 9t2 − 9t − 9)
−(t2 + 3) / (t2 + 8t + 3)
−6(t3 − 9t) / (t3 − 3t2 − 9t + 3)
36
9
[2, 1, 0, 5], [1, 2, 3, 2]
36
9
[2, 1, 0, 5], [4, 0, 3, 5]
9I0 -9c
9J0 -9a
36
9
[2, 2, 0, 5], [2, 2, 3, 1]
36
9
[1, 3, 0, 1], [2, 2, 3, 8], [1, 2, 0, 2]
36
9
[1, 3, 0, 1], [2, 2, 3, 8], [2, 1, 0, 1]
20
9J0 -9b
9J0 -9c
36
9
21
27A0 -27a
36
27
16
17
18
19
3
−9(t3 + 3t2 − 9t − 3)/(8t3 )
37 (t2 −1)3 (t6 +3t5 +6t4 +t3 −3t2 +12t+16)3
(2t3 +3t2 −3t−5)−1 (t3 −3t−1)9
(t3 −9t−12)(9−3t3 )(5t3 +18t2 +18t+3)
(t3 +3t2 −3)3
9I0 -9a
9I0 -9b
15
3
12
10
14
sup
0
1
9
4
4
10
6
−3(t3 + 9t2 − 9t − 9) / (t3 + 3t2 − 9t − 3)
(t3 − 6t2 + 3t + 1) / (t2 − t)
6
7
[1, 3, 0, 1], [5, 2, 3, 5], [4, 0, 0, 5]
(t3 − 3t + 1) / (t2 − t)
−18(t2 − 1) / (t3 − 3t2 − 9t + 3)
3(t3 + 3t2 − 9t − 3) / (t3 − 3t2 − 9t + 3)
[1, 1, 0, 1], [2, 1, 9, 5], [1, 2, 3, 2]
t3
6
6
7
7
group
1A0 -1a
5A0 -5a
5B0 -5a
5C0 -5a
5D0 -5a
5D0 -5b
5E0 -5a
5G0 -5a
i
N
generators
map
supergroup
1
5
6
10
12
12
15
30
1
5
5
5
5
5
5
5
[2, 1, 0, 3], [1, 2, 2, 0], [1, 1, 0, 2]
[2, 0, 0, 3], [1, 0, 1, 1], [1, 0, 0, 2]
[3, 1, 0, 2], [1, 2, 2, 0], [2, 2, 2, 1]
[4, 0, 1, 4], [1, 0, 0, 2]
[4, 0, 1, 4], [2, 0, 0, 1]
[2, 1, 0, 3], [2, 0, 2, 3], [1, 0, 2, 2]
[3, 1, 0, 2], [2, 1, 0, 1]
t3 (t2 + 5t + 40)
(t2 + 10t + 5)3 / t
8000t3 (t + 1)(t2 − 5t + 10)3 / (t2 − 5)5
125t / (t2 − 11t − 1)
(t2 − 11t − 1) / t
(t + 5)(t2 − 5) / (t2 + 5t + 5)
125 / (t(t4 + 5t3 + 15t2 + 25t + 25))
(t2 + 5)/t
−t(t2 + 5t + 10) / (t3 + 5t2 + 10t + 10)
−5(t2 + 4t + 5)/(t2 + 5t + 5)
−1 / t5
−(t4 −2t3 +4t2 −3t+1)
t(t4 +3t3 +4t2 +2t+1)
5t / (t2 − t − 1)
(t − 1)(t4 + t3 + 6t2 + 6t + 11)
−t5
(1 − t2 ) / t
−(t4 −2t3 +4t2 −3t+1)
t(t4 +3t3 +4t2 +2t+1)
(t2 + 4t − 1) / (t2 − t − 1)
(t2 + 5t + 1)3 (t2 + 13t + 49) / t
(2t−1)3 (t2 −t+2)3 (2t2 +5t+4)3 (5t2 +2t−4)3
(t3 +2t2 −t−1)7
49(t2 − t) / (t3 − 8t2 + 5t + 1)
(t3 − 8t2 + 5t + 1) / (t2 − t)
−7(t3 − 2t2 − t + 1) / (t3 − t2 − 2t + 1)
t(t+1)3 (t2 −5t+1)3 (t2 −5t+8)3
(t4 −5t3 +8t2 −7t+7)−3 (t3 −4t2 +3t+1)7
2
(t + 5t + 13)(t4 + 7t3 + 20t2 + 19t + 1)3 / t
13t / (t2 − 3t − 1)
(t2 − 3t − 1) / t
13(t2 − t) / (t3 − 4t2 + t + 1)
(t3 − 4t2 + t + 1) / (t2 − t)
−(5t3 − 7t2 − 8t + 5) / (t3 − 4t2 + t + 1)
1A0 -1a
1A0 -1a
1A0 -1a
5B0 -5a
5B0 -5a
5A0 -5a
5E0 -5a
5B0 -5a
5C0 -5a
5E0 -5a
5D0 -5a
5G0 -5b
30
5
[3, 1, 0, 2], [2, 1, 3, 3]
5H0 -5a
60
5
[4, 0, 0, 4], [2, 0, 0, 1]
25A0 -25a
25B0 -25a
30
60
25
25
[2, 2, 0, 13], [4, 1, 3, 1], [2, 3, 0, 6]
[9, 10, 0, 14], [0, 7, 7, 2], [2, 8, 0, 1]
25B0 -25b
60
25
[9, 10, 0, 14], [0, 7, 7, 2], [4, 1, 0, 7]
7B0 -7a
8
7
[2, 0, 0, 4], [3, 0, 1, 5], [1, 0, 0, 3]
7D0 -7a
21
7
[0, 3, 2, 3], [2, 4, 4, 5], [3, 1, 0, 4]
7E0 -7a
7E0 -7b
7E0 -7c
24
24
24
7
7
7
[6, 0, 1, 6], [1, 0, 0, 3]
[6, 0, 1, 6], [3, 0, 0, 1]
[6, 0, 1, 6], [3, 0, 0, 4]
7F0 -7a
28
7
[3, 1, 4, 4], [4, 4, 1, 3], [3, 4, 0, 4]
13A0 -13a
13B0 -13a
13B0 -13b
13C0 -13a
13C0 -13b
13C0 -13c
14
28
28
42
42
42
13
13
13
13
13
13
[2, 0, 0, 7], [1, 0, 1, 1], [1, 0, 0, 2]
[3, 0, 0, 9], [4, 0, 1, 10], [1, 0, 0, 2]
[3, 0, 0, 9], [4, 0, 1, 10], [2, 0, 0, 1]
[5, 0, 0, 8], [1, 0, 1, 1], [1, 0, 0, 2]
[5, 0, 0, 8], [1, 0, 1, 1], [2, 0, 0, 1]
[5, 0, 0, 8], [1, 0, 1, 1], [2, 0, 0, 3]
5D0 -5b
5G0 -5a
5B0 -5a
5D0 -5b
25A0 -25a
5D0 -5a
25A0 -25a
1A0 -1a
1A0 -1a
7B0 -7a
7B0 -7a
7B0 -7a
1A0 -1a
1A0 -1a
13A0 -13a
13A0 -13a
13A0 -13a
13A0 -13a
13A0 -13a
group
1A0 -1a
2A0 -2a
2A0 -4a
2A0 -8a
2A0 -8b
2B0 -2a
2C0 -2a
2C0 -4a
2C0 -8a
2C0 -8b
4A0 -4a
4B0 -4a
4B0 -4b
4B0 -8a
4B0 -8b
4C0 -4a
4C0 -4b
4C0 -8a
4C0 -8b
4D0 -4a
4D0 -8a
4E0 -4a
4E0 -4b
4E0 -4c
4E0 -8a
4E0 -8b
4E0 -8c
4E0 -8d
4E0 -8e
4E0 -8f
4E0 -8g
4E0 -8h
4E0 -8i
i
N
1
2
2
2
2
3
6
6
6
6
4
6
6
6
6
6
6
6
6
8
8
12
12
12
12
12
12
12
12
12
12
12
12
1
2
4
8
8
2
2
4
8
8
4
4
4
8
8
4
4
8
8
4
8
4
4
4
8
8
8
8
8
8
8
8
8
generators
map
supergroup
[0, 1, 1, 1]
[1, 2, 0, 1], [1, 1, 1, 2], [1, 1, 0, 3]
[1, 2, 0, 1], [1, 1, 1, 2], [1, 0, 0, 3], [1, 1, 0, 5]
[1, 2, 0, 1], [1, 1, 1, 2], [1, 1, 0, 3], [1, 1, 0, 5]
[0, 1, 1, 0]
t2 + 1728
−t2 + 1728
−2t2 + 1728
2t2 + 1728
(256 − t)3 /t2
−t2 + 64
64(t2 + 1)
32(t2 + 2)
−32(t2 − 2)
4t3 (8 − t)
256/(t2 + 4)
−4096/(t2 + 16t)
−512/(t2 − 8)
512/(t2 + 8)
t2
−t2
−128t2
128t2
−(t2 + 2t − 2)/t
(t2 + 4t − 2)/(4 − 2t2 )
(t2 − 1)/(2t)
8(2t2 + 1)
4(t2 + 1)/t
4t/(t2 − 2)
8(t2 − 1)
(t2 − 2)/(2t)
8(t2 + 1)
(t2 + 1)/(t2 − 2t − 1)
(t2 + 2t − 1)/(t2 + 1)
4t/(t2 + 2)
2(t2 + 1)/(t2 + 2t − 1)
(t2 + 2)/(2t)
1A0 -1a
1A0 -1a
1A0 -1a
1A0 -1a
1A0 -1a
2B0 -2a
2B0 -2a
2B0 -2a
2B0 -2a
1A0 -1a
2B0 -2a
2B0 -2a
2B0 -2a
2B0 -2a
2B0 -2a
2B0 -2a
2B0 -2a
2B0 -2a
4A0 -4a
4A0 -4a
2C0 -4a
2C0 -2a
2C0 -2a
2C0 -8a
2C0 -2a
2C0 -8a
2C0 -2a
4C0 -8b
2C0 -8b
2C0 -8b
2C0 -8b
2C0 -8b
[1, 2, 0, 1], [3, 0, 0, 3], [1, 0, 2, 1], [1, 1, 0, 3]
[1, 2, 0, 1], [3, 0, 0, 3], [1, 0, 2, 1], [1, 0, 0, 3], [0, 1, 1, 0]
[1, 2, 0, 1], [3, 0, 0, 3], [1, 0, 2, 1], [1, 1, 0, 3], [1, 1, 0, 5]
[1, 1, 1, 2], [0, 1, 3, 0], [1, 1, 0, 3]
[3, 0, 0, 3], [0, 1, 3, 2], [1, 0, 0, 3]
[3, 0, 0, 3], [0, 1, 3, 2], [1, 2, 0, 3]
[3, 0, 0, 3], [1, 4, 0, 1], [0, 3, 5, 2], [1, 0, 0, 3], [1, 2, 0, 5]
[3, 0, 0, 3], [1, 4, 0, 1], [0, 3, 5, 2], [1, 2, 0, 3], [1, 2, 0, 5]
[1, 2, 2, 1], [0, 1, 3, 0], [1, 0, 0, 3]
[1, 2, 2, 1], [0, 1, 3, 0], [1, 2, 0, 3]
[1, 4, 0, 1], [2, 1, 3, 2], [0, 3, 5, 0], [1, 0, 0, 3], [1, 2, 0, 5]
[1, 4, 0, 1], [2, 1, 3, 2], [0, 3, 5, 0], [1, 2, 0, 3], [1, 2, 0, 5]
[2, 1, 1, 3], [1, 1, 0, 3]
[1, 4, 0, 1], [2, 1, 5, 3], [3, 3, 0, 5], [0, 1, 3, 0]
[3, 0, 0, 3], [1, 2, 2, 1], [0, 1, 1, 0]
[3, 0, 0, 3], [1, 2, 2, 1], [1, 0, 0, 3]
[3, 0, 0, 3], [1, 2, 2, 1], [1, 2, 0, 3]
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 2, 2, 5], [1, 0, 0, 3], [0, 1, 1, 0]
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 2, 2, 5], [1, 0, 0, 3], [1, 2, 0, 5]
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 2, 2, 5], [1, 2, 0, 3], [0, 1, 1, 0]
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 2, 2, 5], [1, 2, 0, 3], [1, 2, 0, 5]
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 2, 2, 5], [1, 2, 0, 5], [0, 1, 1, 0]
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 2, 2, 5], [3, 0, 0, 5], [0, 1, 3, 2]
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 2, 2, 5], [3, 0, 0, 5], [2, 1, 1, 2]
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 2, 2, 5], [3, 2, 0, 5], [0, 1, 3, 0]
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 2, 2, 5], [3, 2, 0, 5], [2, 1, 1, 2]
group
4F0 -4a
4F0 -4b
4F0 -8a
4F0 -8b
4G0 -16a
4G0 -4a
4G0 -4b
4G0 -8a
4G0 -8b
4G0 -8c
4G0 -8d
4G0 -8e
4G0 -8f
8B0 -8a
8B0 -8b
8B0 -8c
8B0 -8d
8C0 -8a
8C0 -8b
8C0 -8c
8C0 -8d
8D0 -8a
8D0 -8b
8D0 -8c
8D0 -8d
8E0 -16a
8E0 -16b
8F0 -8a
8G0 -16a
i
N
generators
12
12
12
12
24
24
24
24
24
24
24
24
24
12
12
12
12
12
12
12
12
12
12
12
12
16
16
16
24
4
4
8
8
16
4
4
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
16
16
8
16
[0, 1, 3, 0], [1, 0, 0, 3]
8(t2 − 1)
[0, 1, 3, 0], [2, 1, 1, 2]
8(t2 + 1)
[3, 0, 0, 3], [1, 4, 0, 1], [0, 3, 5, 0], [1, 0, 0, 3], [1, 2, 2, 1]
4(t2 + 2)
[3, 0, 0, 3], [1, 4, 0, 1], [0, 3, 5, 0], [3, 0, 0, 5], [1, 2, 2, 1]
4(t2 − 2)
[1, 4, 0, 1], [7, 0, 0, 7], [3, 0, 0, 11], [1, 0, 4, 1], [1, 1, 0, 5], [1, 5, 2, 5] (1 − t2 )/(2t)
[3, 0, 0, 3], [1, 0, 0, 3]
1/4t2
[3, 0, 0, 3], [1, 3, 0, 3]
t2 /2
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 0, 4, 1], [0, 1, 3, 0], [2, 1, 5, 2]
4t/(t2 − 2)
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 0, 4, 1], [1, 0, 0, 3], [0, 1, 1, 0]
2(t2 + 2t + 2)/(t2 − 2)
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 0, 4, 1], [1, 0, 0, 3], [1, 2, 0, 5]
t2 /2
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 0, 4, 1], [1, 2, 0, 5], [3, 0, 2, 5]
2(t2 − 1)/(t2 + 1)
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 0, 4, 1], [1, 3, 0, 3], [1, 2, 0, 5]
t2
[3, 0, 0, 3], [1, 4, 0, 1], [5, 0, 0, 5], [1, 0, 4, 1], [1, 3, 0, 3], [1, 3, 2, 3]
4t/(t2 + 2)
[3, 0, 0, 3], [0, 3, 5, 0], [1, 2, 2, 5], [1, 0, 0, 3], [1, 0, 0, 5]
16t2
[3, 0, 0, 3], [0, 3, 5, 0], [1, 2, 2, 5], [1, 0, 0, 3], [1, 4, 0, 5]
32t2
[3, 0, 0, 3], [0, 3, 5, 0], [1, 2, 2, 5], [3, 2, 0, 1], [1, 0, 0, 5]
32t2
[3, 0, 0, 3], [0, 3, 5, 0], [1, 2, 2, 5], [3, 2, 0, 1], [1, 4, 0, 5]
16t2
[3, 0, 0, 3], [5, 0, 0, 5], [0, 3, 5, 2], [1, 2, 0, 3], [1, 0, 0, 5]
−8t2
[3, 0, 0, 3], [5, 0, 0, 5], [0, 3, 5, 2], [1, 2, 0, 3], [1, 4, 0, 5]
−4(t2 + 4)
[3, 0, 0, 3], [5, 0, 0, 5], [0, 3, 5, 2], [3, 2, 0, 1], [1, 0, 0, 5]
−8(t2 + 2)
[3, 0, 0, 3], [5, 0, 0, 5], [0, 3, 5, 2], [3, 2, 0, 1], [1, 4, 0, 5]
−t2
[2, 1, 3, 2], [0, 3, 5, 0], [1, 0, 0, 3], [1, 0, 0, 5]
16/(t2 − 2)
[2, 1, 3, 2], [0, 3, 5, 0], [1, 0, 0, 3], [1, 4, 0, 5]
32/(t2 + 4)
[2, 1, 3, 2], [0, 3, 5, 0], [1, 4, 0, 3], [1, 0, 0, 5]
16/(t2 + 2)
[2, 1, 3, 2], [0, 3, 5, 0], [1, 4, 0, 3], [1, 4, 0, 5]
32/(t2 − 4)
[3, 4, 0, 11], [2, 3, 3, 5], [3, 3, 0, 5], [0, 1, 3, 0]
−4t/(t2 + 2)
[3, 4, 0, 11], [2, 3, 3, 5], [3, 3, 0, 5], [0, 3, 1, 0]
−2(t2 − 2t + 2)/(t2 − 4t + 2)
[1, 1, 1, 2], [0, 3, 5, 0], [3, 3, 0, 5], [2, 1, 1, 3]
8(t4 − 4t2 − 8t − 4)/(t2 − 2)2
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 0, 11], [1, 0, 8, 1], [3, 1, 0, 5], [1, 2, 2, 1] (1 − t2 )/(2t)
map
super
4C0 -4a
4C0 -4a
4C0 -4a
4C0 -4a
4E0 -8f
4E0 -4c
4E0 -4a
4F0 -4b
4F0 -4a
4E0 -4c
4F0 -8b
4E0 -4a
4F0 -4a
4C0 -4a
4C0 -4a
4C0 -4b
4C0 -4b
4B0 -4b
4B0 -4b
4B0 -4b
4B0 -4b
4C0 -4a
4C0 -4a
4C0 -4a
4C0 -4a
4D0 -8a
4D0 -8a
4A0 -4a
4E0 -8e
group
8G0 -8a
8G0 -8b
8G0 -8c
8G0 -8d
8G0 -8e
8G0 -8f
8G0 -8g
8G0 -8h
8G0 -8i
8G0 -8j
8G0 -8k
8G0 -8l
8H0 -8a
8H0 -8b
8H0 -8c
8H0 -8d
8H0 -8e
8H0 -8f
8H0 -8g
8H0 -8h
8H0 -8i
8H0 -8j
8H0 -8k
8H0 -8l
8I0 -8a
8I0 -8b
8I0 -8c
8I0 -8d
8J0 -8a
8J0 -8b
8J0 -8c
8J0 -8d
i
N
generators
map
supergroup
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [1, 0, 0, 3], [1, 0, 0, 5]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [1, 0, 0, 3], [1, 0, 4, 5]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [1, 0, 0, 3], [1, 1, 0, 5]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [1, 0, 0, 5], [3, 0, 2, 1]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [1, 0, 2, 3], [3, 2, 2, 1]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [1, 1, 0, 3], [1, 0, 0, 5]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [1, 1, 0, 3], [1, 1, 4, 1]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [1, 1, 0, 3], [3, 2, 2, 1]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [1, 3, 2, 3], [3, 2, 2, 1]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [3, 0, 0, 5], [1, 1, 4, 1]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [3, 1, 0, 5], [1, 0, 4, 5]
[1, 2, 0, 1], [3, 0, 0, 3], [5, 0, 0, 5], [3, 1, 0, 5], [3, 0, 2, 1]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [1, 0, 0, 3], [1, 0, 0, 5]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [1, 0, 0, 3], [1, 2, 2, 3]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [1, 0, 0, 3], [1, 4, 0, 5]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [1, 0, 0, 5], [1, 2, 2, 3]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [1, 4, 0, 3], [1, 0, 0, 5]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [1, 4, 0, 3], [1, 2, 2, 1]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [1, 4, 0, 3], [1, 4, 0, 5]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [1, 4, 0, 5], [2, 1, 1, 2]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [3, 0, 0, 5], [1, 2, 2, 1]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [3, 0, 0, 5], [2, 3, 5, 2]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [3, 4, 0, 5], [1, 2, 2, 1]
[3, 0, 0, 3], [1, 4, 4, 1], [0, 3, 5, 0], [3, 4, 0, 5], [2, 1, 1, 2]
[7, 0, 0, 7], [0, 3, 5, 2], [1, 4, 0, 5], [1, 6, 0, 3]
[7, 0, 0, 7], [0, 3, 5, 2], [3, 2, 0, 1], [1, 4, 0, 5]
[7, 0, 0, 7], [0, 3, 5, 2], [3, 2, 0, 1], [5, 4, 0, 1]
[7, 0, 0, 7], [0, 3, 5, 2], [5, 2, 0, 3], [5, 4, 0, 1]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 2, 4, 1], [1, 0, 0, 3], [1, 0, 0, 5]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 2, 4, 1], [1, 0, 0, 3], [1, 2, 0, 5]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 2, 4, 1], [1, 2, 0, 3], [1, 0, 0, 5]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 2, 4, 1], [1, 2, 0, 3], [1, 2, 0, 5]
(t2 − 1)/t
(t2 + 2)/(2t)
(t2 − 2)/(2t)
t2 /2
t2
(t2 − 1)/(2t)
(t2 + 2)/(2t)
2(t2 + 1)/(t2 + 2t − 1)
(t2 + 2)/(2t)
2(t2 + 1)/(t2 − 2t − 1)
2(t2 − 1)/(t2 + 1)
(t2 − 2)/(2t)
2t/(t2 − 2)
4t/(t2 − 2)
t/(t2 + 1)
(t2 + 1)/(t2 + 2t − 1)
2t/(t2 + 2)
(t2 − 2)/(2t)
t/(t2 − 1)
(t2 − 1)/(2t)
(t2 + 2)/(2t)
2(t2 + 1)/(t2 + 2t − 1)
(t2 + 2t − 1)/(t2 + 1)
4t/(t2 + 2)
4t2 /(t2 − 2)
4t2 /(t2 + 2)
4/(t2 − 1)
4/(t2 + 1)
(t2 + 2)/t2
t2 − 1
t2 /(t2 − 2)
t2 + 1
8C0 -8b
8C0 -8a
4E0 -8c
4E0 -4b
4E0 -4b
4E0 -4a
4E0 -8i
4E0 -8g
4E0 -8g
4E0 -8i
8D0 -8a
4E0 -8a
8B0 -8a
4F0 -8a
8B0 -8b
8B0 -8a
8B0 -8a
4F0 -8a
8B0 -8b
4F0 -4b
4F0 -8b
4F0 -8b
4F0 -8b
4F0 -8b
8C0 -8d
8C0 -8d
8C0 -8d
8C0 -8d
4E0 -4c
4E0 -4c
4E0 -4c
4E0 -4c
group
8K0 -16a
8K0 -16b
8K0 -16c
8K0 -16d
8L0 -8a
8L0 -8b
8N0 -16a
8N0 -16b
8N0 -16c
8N0 -16d
8N0 -16e
8N0 -16f
8N0 -32a
8N0 -8a
8N0 -8b
8N0 -8c
8N0 -8d
8N0 -8e
8N0 -8f
8O0 -16a
8O0 -8a
8O0 -8b
8O0 -8c
8O0 -8d
8O0 -8e
8O0 -8f
8O0 -8g
8O0 -8h
8O0 -8i
8O0 -8j
8O0 -8k
8O0 -8l
8P0 -8a
8P0 -8b
i
N
generators
map
super
24
24
24
24
24
24
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
16
16
16
16
8
8
16
16
16
16
16
16
32
8
8
8
8
8
8
16
8
8
8
8
8
8
8
8
8
8
8
8
8
8
[1, 4, 0, 1], [7, 0, 0, 7], [0, 3, 5, 0], [3, 0, 0, 5], [1, 2, 2, 1]
[1, 4, 0, 1], [7, 0, 0, 7], [0, 3, 5, 0], [7, 0, 0, 9], [1, 2, 2, 1]
[1, 4, 0, 1], [7, 0, 0, 7], [2, 3, 1, 2], [3, 0, 0, 5], [1, 2, 2, 1]
[1, 4, 0, 1], [7, 0, 0, 7], [2, 3, 1, 2], [7, 0, 0, 9], [1, 2, 2, 1]
[0, 3, 5, 0], [5, 2, 2, 1], [1, 2, 0, 3], [1, 4, 0, 5]
[0, 3, 5, 0], [5, 2, 2, 1], [5, 4, 0, 1], [3, 6, 0, 1]
[1, 4, 0, 1], [7, 0, 0, 7], [9, 0, 0, 9], [1, 0, 4, 1], [0, 1, 1, 0], [2, 3, 5, 6]
[1, 4, 0, 1], [7, 0, 0, 7], [9, 0, 0, 9], [1, 0, 4, 1], [0, 1, 5, 0], [2, 5, 1, 2]
[1, 4, 0, 1], [7, 0, 0, 7], [9, 0, 0, 9], [1, 0, 4, 1], [1, 2, 2, 1], [0, 5, 1, 0]
[1, 4, 0, 1], [7, 0, 0, 7], [9, 0, 0, 9], [1, 0, 4, 1], [3, 2, 2, 5], [2, 3, 5, 6]
[1, 4, 0, 1], [7, 0, 0, 7], [9, 0, 0, 9], [1, 0, 4, 1], [7, 0, 0, 9], [1, 2, 2, 1]
[1, 4, 0, 1], [7, 0, 0, 7], [9, 0, 0, 9], [1, 0, 4, 1], [7, 0, 0, 9], [2, 1, 1, 6]
[1, 4, 0, 1], [15, 0, 0, 15], [7, 0, 0, 23], [1, 0, 4, 1], [3, 0, 0, 5], [2, 3, 1, 0]
[1, 4, 0, 1], [7, 0, 0, 7], [1, 0, 4, 1], [1, 2, 0, 3], [1, 0, 0, 5]
[1, 4, 0, 1], [7, 0, 0, 7], [1, 0, 4, 1], [1, 2, 0, 3], [1, 2, 0, 5]
[1, 4, 0, 1], [7, 0, 0, 7], [1, 0, 4, 1], [1, 2, 0, 3], [3, 2, 0, 5]
[1, 4, 0, 1], [7, 0, 0, 7], [1, 0, 4, 1], [3, 0, 0, 1], [1, 0, 2, 5]
[1, 4, 0, 1], [7, 0, 0, 7], [1, 0, 4, 1], [3, 0, 0, 1], [1, 2, 0, 5]
[1, 4, 0, 1], [7, 0, 0, 7], [1, 0, 4, 1], [3, 0, 0, 1], [5, 0, 0, 1]
[1, 4, 0, 1], [7, 0, 0, 7], [3, 2, 0, 11], [1, 0, 8, 1], [1, 3, 0, 5], [5, 1, 4, 3]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 0, 0, 3], [1, 0, 4, 5]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 0, 0, 5], [1, 0, 4, 3]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 2, 0, 3], [1, 0, 0, 5]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 2, 0, 3], [1, 2, 0, 5]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 2, 0, 5], [1, 2, 4, 3]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 3, 0, 3], [1, 0, 0, 5]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 3, 0, 3], [1, 2, 0, 5]
[3, 2, 0, 3], [5, 2, 0, 5], [1, 3, 0, 3], [1, 3, 4, 1]
[3, 2, 0, 3], [5, 2, 0, 5], [3, 0, 0, 5], [1, 2, 4, 3]
[3, 2, 0, 3], [5, 2, 0, 5], [3, 2, 0, 5], [1, 0, 4, 3]
[3, 2, 0, 3], [5, 2, 0, 5], [3, 3, 0, 5], [1, 0, 4, 3]
[3, 2, 0, 3], [5, 2, 0, 5], [3, 3, 0, 5], [1, 1, 4, 1]
[3, 4, 4, 3], [0, 3, 5, 0], [3, 4, 0, 1], [1, 4, 0, 5]
[3, 4, 4, 3], [0, 3, 5, 0], [3, 4, 0, 1], [5, 4, 0, 1]
(t2 + 4)/2
(t2 − 4)/2
t2 + 2
t2 − 2
(t2 − 2t − 1)/(4t)
2t/(t2 − 2t − 1)
(t2 + 2)/(2t)
t2 /2
4t/(t2 − 1)
2(t2 + 1)/(t2 − 2t − 1)
(t2 − 1)/(2t)
(t2 − 2)/(2t)
(1 − 2t − t2 )/(t2 − 2t − 1)
t2
(t2 − 2)/(2t)
(t2 + 1)/t
2(t2 + 1)/(t2 + 2t − 1)
(t2 + 2)/(2t)
(t2 − 1)/t
(1 − t2 )/(2t)
(t2 − 2)/(2t)
(t2 − 1)/(2t)
2t/(t2 − 1)
(t2 + 2)/(2t)
(t2 + 1)/(2t)
t2 /2
t2
(t2 − 2)/(2t)
(t2 − 2)/(t2 − 4t + 2)
(t2 + 2)/(2t)
(t2 + 4t + 2)/(t2 − 2)
(t2 + 2)/(2t)
(t2 + 2)/(t2 − 4t + 2)
(t2 + 1)/(2(t − 1))
4F0 -8b
4F0 -8b
4F0 -8b
4F0 -8b
8B0 -8d
8B0 -8d
4G0 -8a
4G0 -8e
8K0 -16c
4G0 -8a
4G0 -8d
4G0 -8f
4G0 -16a
4G0 -4a
4G0 -8c
4G0 -4a
4G0 -8c
4G0 -8c
4G0 -4a
8G0 -8j
8G0 -8b
8G0 -8a
8J0 -8a
8J0 -8b
8I0 -8a
8G0 -8f
8G0 -8f
8G0 -8g
8J0 -8a
8G0 -8b
8I0 -8b
8G0 -8c
8H0 -8g
8H0 -8g
group
16B0 -16a
16B0 -16b
16B0 -16c
16B0 -16d
16C0 -16a
16C0 -16b
16C0 -16c
16C0 -16d
16D0 -16a
16D0 -16b
16D0 -16c
16D0 -16d
16E0 -16a
16E0 -16b
16E0 -16c
16E0 -16d
16F0 -32a
16F0 -32b
16G0 -16a
16G0 -16b
16G0 -16c
16G0 -16d
16G0 -16e
16G0 -16f
16G0 -16g
16G0 -16h
16G0 -16i
16G0 -16j
16G0 -16k
16G0 -16l
16G0 -32a
i
N
generators
map
supergroup
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
32
32
48
48
48
48
48
48
48
48
48
48
48
48
48
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
32
32
16
16
16
16
16
16
16
16
16
16
16
16
32
[3, 0, 0, 11], [0, 3, 5, 0], [1, 2, 2, 5], [1, 0, 0, 5], [1, 8, 0, 3]
[3, 0, 0, 11], [0, 3, 5, 0], [1, 2, 2, 5], [1, 4, 0, 5], [5, 2, 0, 3]
[3, 0, 0, 11], [0, 3, 5, 0], [1, 2, 2, 5], [3, 0, 0, 5], [1, 8, 0, 3]
[3, 0, 0, 11], [0, 3, 5, 0], [1, 2, 2, 5], [3, 6, 0, 5], [3, 4, 0, 7]
[7, 0, 0, 7], [3, 8, 0, 11], [0, 3, 5, 2], [1, 2, 0, 3], [1, 4, 0, 5]
[7, 0, 0, 7], [3, 8, 0, 11], [0, 3, 5, 2], [1, 2, 0, 3], [5, 4, 0, 1]
[7, 0, 0, 7], [3, 8, 0, 11], [0, 3, 5, 2], [1, 4, 0, 5], [5, 2, 0, 3]
[7, 0, 0, 7], [3, 8, 0, 11], [0, 3, 5, 2], [5, 4, 0, 1], [1, 6, 0, 3]
[7, 0, 0, 7], [3, 0, 0, 11], [0, 3, 5, 2], [1, 4, 0, 5], [5, 2, 0, 3]
[7, 0, 0, 7], [3, 0, 0, 11], [0, 3, 5, 2], [3, 2, 0, 1], [1, 4, 0, 5]
[7, 0, 0, 7], [3, 0, 0, 11], [0, 3, 5, 2], [3, 2, 0, 1], [5, 2, 0, 3]
[7, 0, 0, 7], [3, 0, 0, 11], [0, 3, 5, 2], [3, 6, 0, 5], [3, 4, 0, 7]
[0, 3, 5, 8], [2, 1, 3, 10], [1, 0, 0, 3], [1, 0, 0, 5]
[0, 3, 5, 8], [2, 1, 3, 10], [1, 0, 0, 3], [1, 8, 0, 5]
[0, 3, 5, 8], [2, 1, 3, 10], [1, 0, 0, 5], [1, 8, 0, 3]
[0, 3, 5, 8], [2, 1, 3, 10], [3, 0, 0, 5], [1, 8, 0, 3]
[3, 4, 0, 11], [6, 3, 7, 9], [3, 3, 0, 5], [0, 3, 1, 0]
[3, 4, 0, 11], [6, 3, 7, 9], [5, 5, 0, 3], [0, 3, 1, 0]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [1, 3, 2, 3], [3, 2, 2, 1]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [3, 0, 0, 1], [1, 0, 0, 5]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [3, 0, 0, 1], [5, 0, 0, 1]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [3, 0, 0, 1], [5, 1, 0, 1]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [3, 1, 0, 1], [1, 1, 0, 5]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [3, 1, 0, 1], [1, 1, 4, 1]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [3, 1, 0, 1], [5, 0, 0, 1]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [3, 2, 2, 1], [1, 4, 2, 3]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [3, 2, 2, 1], [3, 3, 4, 5]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [3, 4, 2, 1], [1, 2, 4, 5]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [7, 1, 0, 9], [1, 2, 4, 5]
[1, 2, 0, 1], [7, 0, 0, 7], [3, 0, 8, 11], [7, 1, 0, 9], [3, 0, 2, 1]
[1, 2, 0, 1], [15, 0, 0, 15], [7, 0, 0, 23],
[3, 0, 8, 11], [3, 1, 0, 5], [5, 2, 2, 5]
t2 /2
t2 /2
t2
t2
t2 /2
t2
2t2
t2
t2 − 4
t2 + 4
2(t2 − 2)
2(t2 + 2)
(t2 + 4)/2
t2 − 2
(t2 − 4)/2
t2 + 2
−(t2 + 2)/(2t)
(t2 − 4t + 2)/(t2 − 2t + 2)
2(t2 + 1)/(t2 + 2t − 1)
t2 /2
t2
(t2 − 2)/(2t)
(t2 + 2)/(2t)
2(t2 + 1)/(t2 + 2t − 1)
(t2 − 1)/(2t)
t2
(t2 + 2)/(2t)
t2 /2
(t2 − 1)/(2t)
(t2 − 2)/(2t)
(1 − 2t − t2 )/(t2 − 2t − 1)
8B0 -8a
8B0 -8d
8B0 -8a
8B0 -8d
8C0 -8b
8C0 -8b
8C0 -8d
8C0 -8d
8C0 -8d
8C0 -8d
8C0 -8d
8C0 -8d
8D0 -8a
8D0 -8a
8D0 -8a
8D0 -8a
8E0 -16b
8E0 -16b
8G0 -8i
8G0 -8a
8G0 -8a
8G0 -8c
8G0 -8g
8G0 -8g
8G0 -8f
8G0 -8e
8G0 -8i
8G0 -8e
8G0 -8k
8G0 -8l
8G0 -16a
group
16H0 -16a
16H0 -16b
16H0 -16c
16H0 -16d
16H0 -16e
16H0 -16f
16H0 -16g
16H0 -16h
16H0 -16i
16H0 -16j
16H0 -16k
16H0 -16l
32A0 -32a
32A0 -32b
32A0 -32c
32A0 -32d
i
N
generators
map
supergroup
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
48
16
16
16
16
16
16
16
16
16
16
16
16
32
32
32
32
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [1, 4, 0, 5], [1, 2, 0, 7]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [1, 4, 0, 5], [1, 6, 0, 3]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [1, 4, 0, 5], [5, 2, 0, 3]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [1, 6, 0, 3], [1, 2, 0, 7]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [3, 2, 0, 1], [1, 2, 0, 7]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [3, 2, 0, 1], [5, 2, 0, 3]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [3, 2, 0, 1], [5, 4, 0, 1]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [3, 4, 0, 7], [5, 6, 0, 7]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [5, 2, 0, 3], [1, 6, 0, 3]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [5, 2, 0, 3], [5, 4, 0, 1]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [5, 4, 0, 1], [1, 2, 0, 7]
[7, 0, 0, 7], [9, 0, 0, 9], [0, 3, 5, 2], [7, 2, 0, 5], [7, 4, 0, 3]
[3, 8, 0, 11], [5, 8, 0, 13], [0, 7, 9, 2], [1, 4, 0, 5], [5, 6, 0, 3]
[3, 8, 0, 11], [5, 8, 0, 13], [0, 7, 9, 2], [5, 2, 0, 3], [5, 4, 0, 1]
[3, 8, 0, 11], [5, 8, 0, 13], [0, 7, 9, 2], [7, 2, 0, 1], [9, 2, 0, 3]
[3, 8, 0, 11], [5, 8, 0, 13], [0, 7, 9, 2], [7, 2, 0, 5], [7, 4, 0, 3]
4t/(t2 + 2)
2(t2 + 1)/(t2 − 2t − 1)
4t/(t2 − 2)
(t2 + 2t − 1)/(t2 + 1)
(t2 − 2)/(2t)
(t2 − 2)/(2t)
(t2 + 1)/t
(t2 + 2)/(2t)
(t2 + 2t − 1)/(t2 + 1)
2t/(t2 − 1)
(t2 − 1)/t
(t2 + 2)/(2t)
2/t2
t2
t2
2/t2
8I0 -8a
8I0 -8a
8I0 -8b
8I0 -8a
16C0 -16c
8I0 -8b
16C0 -16d
16C0 -16c
16C0 -16c
8I0 -8d
16C0 -16d
8I0 -8a
16C0 -16a
16C0 -16d
16C0 -16a
16C0 -16d
group
i
N
generators
curve
map
supergroup
16C1 -16c
16C1 -16d
16B1 -16a
16B1 -16c
16I1 -16d
16I1 -16f
24
24
24
24
48
16
16
16
16
16
[2, 1, 3, 2], [0, 3, 5, 8], [1, 0, 0, 5], [1, 8, 0, 3]
[2, 1, 3, 2], [0, 3, 5, 8], [3, 0, 0, 5], [1, 8, 0, 3]
[3, 0, 0, 11], [0, 3, 5, 0], [2, 3, 9, 6], [1, 0, 0, 3], [1, 0, 0, 5]
[3, 0, 0, 11], [0, 3, 5, 0], [2, 3, 9, 6], [1, 4, 0, 3], [1, 0, 0, 5]
[3, 0, 0, 11], [0, 3, 5, 8], [1, 4, 12, 1], [1, 0, 0, 5], [1, 8, 0, 3]
256a2
256a1
256b2
256b1
256a1
48
16
[3, 0, 0, 11], [0, 3, 5, 8], [1, 4, 12, 1], [1, 8, 0, 3], [2, 3, 5, 2]
256a2
8D0 -8a
8D0 -8a
8B0 -8a
8B0 -8a
8H0 -8a
8H0 -8b
16I1 -16g
48
16
[3, 0, 0, 11], [0, 3, 5, 8], [1, 4, 12, 1], [1, 8, 0, 5], [1, 2, 10, 3]
256a2
16I1 -16h
16I1 -16j
16I1 -16k
8H1 -16b
8H1 -16c
48
48
48
16
16
16
[3, 0, 0, 11], [0, 3, 5, 8], [1, 4, 12, 1], [1, 8, 0, 7], [1, 2, 10, 7]
[3, 0, 0, 11], [0, 3, 5, 8], [1, 4, 12, 1], [3, 0, 0, 5], [1, 8, 0, 3]
[3, 0, 0, 11], [0, 3, 5, 8], [1, 4, 12, 1], [3, 0, 0, 5], [2, 3, 5, 2]
256a1
256a2
256a2
48
48
16
16
[7, 0, 0, 7], [1, 8, 0, 1], [1, 4, 4, 1], [0, 3, 5, 0], [3, 0, 0, 5], [1, 2, 2, 9]
[7, 0, 0, 7], [1, 8, 0, 1], [1, 4, 4, 1], [0, 3, 5, 0], [3, 0, 0, 5], [1, 2, 6, 9]
256a2
256a2
8H1 -16e
8H1 -16g
16D1 -16d
8H1 -16j
48
48
24
16
16
16
[7, 0, 0, 7], [1, 8, 0, 1], [1, 4, 4, 1], [0, 3, 5, 0], [7, 0, 0, 9], [1, 2, 2, 1]
[7, 0, 0, 7], [1, 8, 0, 1], [1, 4, 4, 1], [0, 3, 5, 0], [7, 0, 0, 9], [2, 3, 5, 2]
[3, 8, 0, 11], [0, 3, 5, 0], [5, 2, 2, 1], [3, 2, 0, 5], [5, 4, 0, 1]
256a1
256a1
128a1
48
16
[7, 0, 0, 7], [1, 8, 0, 1], [1, 4, 4, 1], [0, 3, 5, 0], [7, 4, 0, 9], [1, 2, 2, 9]
256a2
24
16
[7, 0, 0, 7], [3, 4, 0, 11], [0, 3, 5, 0], [3, 0, 0, 5], [1, 2, 2, 9]
256a1
48
16
[7, 0, 0, 7], [1, 8, 0, 1], [1, 4, 4, 1], [0, 3, 5, 0], [7, 4, 0, 9], [1, 2, 6, 9]
256a2
(x − 1)/2
x+1
x/4
x/2
x−1
4x−4y+12
x2 −2x−15
2(y−x−1)
x2 −2x−11
1/x
(x + 3)/2
−x+1
x+3
(x + 3)/2
x−1
x+3
x−1
−1/x
(x + 1)/2
2(x+y+1)
x2 −2x−11
x+1
4(x−y+3)
x2 −2x−15
(x − 1)/2
x2 +2x−7
8x−8
(1 − x)/2
−2/x
x2 +2y−8
x2 −8x+8
4/x
x
x/2
x2 +2y−8
x2 +4x+8
J11 (x, y)
8D1 -16b
8H1 -16k
8D1 -16c
24
16
[7, 0, 0, 7], [3, 4, 0, 11], [0, 3, 5, 0], [3, 4, 0, 5], [1, 2, 2, 1]
256a2
16J1 -16e
16J1 -16g
16F1 -16a
48
16
[7, 0, 0, 7], [0, 3, 5, 0], [5, 2, 2, 1], [1, 6, 0, 7], [7, 4, 0, 3]
128a2
48
48
16
16
[7, 0, 0, 7], [0, 3, 5, 0], [5, 2, 2, 1], [5, 4, 0, 1], [3, 6, 0, 1]
[3, 0, 0, 11], [1, 4, 4, 1], [0, 3, 5, 0], [1, 0, 0, 3], [1, 0, 0, 5]
128a2
256b1
16F1 -16c
48
16
[3, 0, 0, 11], [1, 4, 4, 1], [0, 3, 5, 0], [1, 0, 0, 5], [1, 2, 2, 3]
256b2
16F1 -16d
16F1 -16h
16F1 -16j
48
48
48
16
16
16
[3, 0, 0, 11], [1, 4, 4, 1], [0, 3, 5, 0], [1, 4, 0, 3], [1, 0, 0, 5]
[3, 0, 0, 11], [1, 4, 4, 1], [0, 3, 5, 0], [3, 0, 0, 5], [1, 2, 2, 1]
[3, 0, 0, 11], [1, 4, 4, 1], [0, 3, 5, 0], [3, 4, 0, 1], [1, 2, 2, 1]
256b2
256b1
256b2
16F1 -16k
48
16
[3, 0, 0, 11], [1, 4, 4, 1], [0, 3, 5, 0], [3, 4, 0, 5], [1, 2, 2, 1]
256b2
11C1 -11a
55
11
[3, 4, 4, 2], [3, 1, 1, 8], [6, 4, 0, 5]
121b1
8H0 -8d
8H0 -8j
8H0 -8a
8H0 -8j
8H0 -8i
8H0 -8j
8H0 -8i
8H0 -8j
8B0 -8d
8H0 -8k
4F0 -8b
8H0 -8l
4F0 -8b
8B0 -8d
8L0 -8b
8H0 -8a
16B0 -16c
8H0 -8e
8H0 -8i
8H0 -8f
16B0 -16a
1A0 -1a
The map J11 (x, y) is due to Halberstadt [Hal98], and is defined by
J11 (x, y) :=
(f1 f2 f3 f4 )3
,
f52 f611
where
f1 := x2 + 3x − 6,
f2 := 11(x2 − 5)y + (2x4 + 23x3 − 72x2 − 28x + 127),
f3 := 6y + 11x − 19,
f4 := 22(x − 2)y + (5x3 + 17x2 − 112x + 120),
f5 := 11y + (2x2 + 17x − 34),
f6 := (x − 4)y − (5x − 9).
References
[Den75] J. B. Dennin Jr., The genus of subfields of K(n), Proc. Amer. Math. Soc. 51
(1975), 282–288.
[Hal98] E. Halberstadt, Sur la courbe modulaire Xndep (11), Experimental Math. 7
(1998), 163–174.
[RZB15] J. Rouse and D. Zureick-Brown, Elliptic curves over Q and 2-adic images of
Galois, Research in Number Theory 1 (2015).
[S15] A. V. Sutherland, Computing images of Galois representations attached to elliptic
curves, Forum of Mathematics, Sigma 4 (2016), 79 pages.
[Z15] D. Zywina, On the possible images of the mod ell representations associated to
elliptic curves over Q, arXiv:1508.07660.
[Z15] D. Zywina, Possible indices for the Galois image of elliptic curves over Q,
arXiv:1508.07663.